A pilot signal or preamble, which is used in a mobile communication system, means a reference signal used for acquisition of initial synchronization, cell detection, channel estimation, etc. An orthogonal or quasi-orthogonal code having good correlation properties may be used as a code sequence which constitutes the preamble.
For example, in case of PI (Portable Internet, Specifications for 2.3 GHz band Portable Internet Service—Physical Layer), 127 sequences excluding one of which components are all 1 are inserted in a frequency domain by using 128×128 Hadamard matrix.
A CAZAC (Constant Amplitude Zero Auto-Correlation) sequence is mainly characterized in that its amplitude is uniform and autocorrelation is represented by a delta function type. However, cross correlation of the CAZAC sequence, although not zero, has a small value. GCL CAZAC sequence and zadoff CAZAC sequence, which are mainly used as the CAZAC sequences, are very similar to each other and have different orientations in phase.
First, the GCL CAZAC sequence is given by the following Equations 1 and 2.
                              c          ⁡                      (                                          k                ;                N                            ,              M                        )                          =                              exp            ⁡                          (                              -                                                      j                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                                          Mk                      ⁡                                              (                                                  k                          +                          1                                                )                                                                              N                                            )                                ⁢                      (                          in              ⁢                                                          ⁢              case              ⁢                                                          ⁢              where              ⁢                                                          ⁢              N              ⁢                                                          ⁢              is              ⁢                                                          ⁢              an              ⁢                                                          ⁢              odd              ⁢                                                          ⁢              number                        )                                              [                  Equation          ⁢                                          ⁢          1                ]                                          c          ⁡                      (                                          k                ;                N                            ,              M                        )                          =                              exp            (                          -                                                j                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                      Mk                    2                                                  N                                      )                    ⁢                      (                          in              ⁢                                                          ⁢              case              ⁢                                                          ⁢              where              ⁢                                                          ⁢              N              ⁢                                                          ⁢              is              ⁢                                                          ⁢              an              ⁢                                                          ⁢              even              ⁢                                                          ⁢              number                        )                                              [                  Equation          ⁢                                          ⁢          2                ]            
The Zadoff CAZAC sequence can be expressed by the following Equations 3 and 4.
                              c          ⁡                      (                                          k                ;                N                            ,              M                        )                          =                              exp            ⁡                          (                                                j                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                      Mk                    ⁡                                          (                                              k                        +                        1                                            )                                                                      N                            )                                ⁢                      (                          in              ⁢                                                          ⁢              case              ⁢                                                          ⁢              where              ⁢                                                          ⁢              N              ⁢                                                          ⁢              is              ⁢                                                          ⁢              an              ⁢                                                          ⁢              odd              ⁢                                                          ⁢              number                        )                                              [                  Equation          ⁢                                          ⁢          3                ]                                          c          ⁡                      (                                          k                ;                N                            ,              M                        )                          =                              exp            (                                          j                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                  Mk                  2                                            N                        )                    ⁢                      (                          in              ⁢                                                          ⁢              case              ⁢                                                          ⁢              where              ⁢                                                          ⁢              N              ⁢                                                          ⁢              is              ⁢                                                          ⁢              an              ⁢                                                          ⁢              even              ⁢                                                          ⁢              number                        )                                              [                  Equation          ⁢                                          ⁢          4                ]            
In the Equations 1 to 4, N is a length of the sequence, M is a natural number which is relatively prime to N among natural numbers less than N, and k represents 0, 1, . . . , N.
Binary Hadamard codes or poly-phase CAZAC codes are orthogonal codes, and the number of the binary Hadamard codes or the poly-phase CAZAC codes, which maintains orthogonality, is limited. The number of orthogonal codes having a length of N, which can be obtained by N×N Hadamard matrix, is N, and the number of orthogonal codes having a length of N, which can be obtained by CAZAC code is equivalent to the number of natural numbers less than N, wherein the natural numbers are relatively prime to N. [David C. Chu, “Polyphase Codes with Good Periodic Correlation Propertie”, Information Theory IEEE Transaction on, vol. 18, issue 4, pp. 531-532, July, 1972]
For example, the length of one OFDM (Orthogonal Frequency Division Multiplexing) symbol in an OFDM system generally has a length of exponentiation of 2 to expedite FFT (Fast Fourier Transform) and IFFT (Inverse Fast Fourier Transform). In this case, if the sequence is generated by the Hadamard code, sequence types equivalent to the total length may be generated. If the sequence is generated by the CAZAC code, sequence types equivalent to N/2 may be generated. For this reason, a problem occurs in that there is limitation in the number of the sequence types.